The outer electrons of large atoms: ionization bounds and universal scales

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Family 263 · Mathematical physics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

A nucleus of charge $Z$ attracts electrons, while the electrons repel one another. How many extra electrons can remain bound? How much energy does it take to remove one electron from a neutral atom? And how far from the nucleus are its outermost electrons?

These questions concern the edge of the electron cloud. They are surprisingly difficult even when the leading total energy of a large atom is understood. The total energy grows on a large scale, while a few outer electrons account for only a small difference between two large numbers.

Family 263 contains three manuscripts claiming uniform ionization bounds and the energy and radius parts of the generalized ionization conjecture. Their model is the full nonrelativistic Coulomb Hamiltonian with two electron spin states. In particular, the claims allow correlated many-electron wave functions, rather than restricting the state to a mean-field approximation.

2. Background and essential definitions

For one fixed nucleus, the $N$-electron Hamiltonian in atomic units is

$$H_{Z,N}=\sum_{i=1}^N\left(-\frac12\Delta_i-\frac{Z}{|x_i|}\right) +\sum_{1\leq i<j\leq N}\frac1{|x_i-x_j|}.$$

The first term is kinetic energy, the second attraction to the nucleus, and the last electron-electron repulsion. The wave function is antisymmetric under exchange of the full position and spin labels of two electrons. This implements the Pauli principle. The nucleus is held fixed, and there are two possible spin states per electron.

Let $E_Z(N)$ be the bottom of the spectrum, the smallest possible energy in the sector with $N$ electrons. Sending an electron arbitrarily far away shows why these energies are nonincreasing with $N$. A sector binds strictly when

$$E_Z(N)<E_Z(N-1).$$

Strict binding is stronger than equality with a threshold, where a ground state might or might not exist.

A neutral atom has $N=Z$. Its energy for removing $m$ electrons is

$$I_m(Z)=E_Z(Z-m)-E_Z(Z).$$

This is the total cost of removing $m$ electrons, not the cost of only the last electron in that sequence.

For a normalized neutral ground state $\Psi_Z$, let $\rho_{\Psi_Z}(x)$ be its spin-summed one-electron density. Its integral is $Z$. Define an outer-electron radius by

$$R_m(\Psi_Z)=\inf\left\{r\geq0: \int_{|x|>r}\rho_{\Psi_Z}(x)\,dx\leq m\right\}.$$

The quantity outside the sphere is an expected electron number. A radius leaving half an electron outside is meaningful even though an actual measurement counts integer electrons. Also, this definition concerns a neutral atom with $Z$ electrons; it does not mean that $m$ electrons have physically been removed.

3. Development history and previous partial results

Thomas and Fermi introduced a statistical approximation replacing the many-electron wave function by a density. Lieb and Simon established its variational theory and leading large-charge energy and bulk-density limits. The total energy scale is $Z^{7/3}$, and the bulk length scale is $Z^{-1/3}$.

Those successes do not settle ionization. If two energy formulas each have a large error, subtracting them need not accurately determine a much smaller difference. Similarly, an outer tail containing one electron represents only a fraction $1/Z$ of a neutral atom. Convergence of the bulk density can miss precisely that tail.

For the full quantum model, Lieb obtained charge bounds linear in nuclear charge. Lieb, Sigal, Simon and Thirring established asymptotic neutrality: the largest strictly bound electron number divided by $Z$ tends to one for large atoms. Subsequent sublinear and explicit estimates improved the excess-charge bound. But a statement such as “the excess is small compared with $Z$” still permits an excess tending slowly to infinity.

Seco, Sigal and Solovej obtained quantitative bounds on first ionization energy and an outer-electron radius for the full model. Their estimates depended on powers of $Z$, leaving uniform bounds distinct. M1 and M2 explain the relevant normalizations and the improved exponent recorded in that earlier paper.

Uniform ionization results were achieved earlier in approximate models, including Thomas–Fermi–von Weizsäcker theory and reduced Hartree–Fock theory. Solovej’s unrestricted Hartree–Fock theorem established bounded excess charge, bounded first ionization energy, and outer-radius control. These results are major predecessors, but their variational spaces differ from the fully correlated Schrödinger model.

Solovej’s A new look at Thomas–Fermi theory formulates generalized ionization laws using upper and lower large-$Z$ limits, followed by a large outer-electron-number limit. The current family claims the corresponding full-model conclusions, with a stronger joint limit for energy but the stated iterated limits for radii.

4. What the new manuscripts claim

Uniform charge, energy and outer radius

M1 first considers a molecule with $M$ fixed nuclei at arbitrary distinct positions, real charges $z_j\geq1$, and total charge $Z=\sum_j z_j$. It claims a universal constant $C$ such that

$$E(N)<E(N-1)\quad\Longrightarrow\quad N\leq Z+CM.$$

The same constant is independent of the nuclear charges and separations. For $N>Z+CM$, adjacent sector energies are equal. This resolves the paper’s strict-binding formulation of the excess-charge conjecture; it does not exclude all possible ground states at threshold.

For a neutral atom with integer $Z\geq1$, M1 additionally claims constants $0<c\leq C<\infty$, independent of $Z$, with

$$c\leq I_1(Z)\leq C, \qquad c\leq R_{1/2}(\Psi_Z)\leq C.$$

The radius statement holds for every normalized neutral ground state. No uniqueness or spherical symmetry of the state is assumed. The positive lower bound on first ionization energy is an additional conclusion beyond the usual boundedness assertion.

Removing many outer electrons

M2 claims the energy asymptotic

$$\frac{I_m(Z)}{m^{7/3}}\longrightarrow a_{\mathrm{TF}}>0$$

whenever the integers satisfy $m\to\infty$ and $Z/m\to\infty$. Thus many electrons are removed, but their number remains negligible compared with the nuclear charge.

The constant is the Thomas–Fermi coefficient for two spin states and kinetic energy $-\Delta/2$. It is characterized using the classical Thomas–Fermi functional, rather than a fitted coefficient. M2 also obtains the conjectured iterated upper and lower limits: take $Z\to\infty$ at fixed $m$, using limsup or liminf, normalize by $m^{7/3}$, and then let $m\to\infty$.

No convergence of $I_m(Z)$ at a fixed $m$ is asserted. In particular, this theorem does not say that the first ionization energies approach one universal number as $Z$ grows.

Radii in neutral atoms

M3 claims, for every choice of neutral ground states,

$$\lim_{m\to\infty}m^{1/3} \left[\limsup_{Z\to\infty}R_m(\Psi_Z)\right] =b_{\mathrm{TF}},$$

and the same formula with liminf, where

$$b_{\mathrm{TF}}=\left(\frac{81\pi^2}{2}\right)^{1/3}.$$

All limits are over integers with $Z>m$. The inner limit keeps $m$ fixed. Neither convergence at fixed $m$ nor an arbitrary joint-limit radius theorem is part of this statement. The result is uniform over the choice of vector in each ground eigenspace.

M2 and M3 use screening and localization estimates supplied by M1. M3 shares an intermediate-scale strategy with M2, but states that its own proof uses the foundation companion and supplies its own neutral-profile and tail arguments.

5. Why the advance matters

Provisional importance: 5/5. Ionization is a central test of how quantum mechanics controls matter beyond its leading bulk behavior. The claimed results describe a small exterior portion of a potentially enormous electron cloud, with constants that do not grow with nuclear charge.

The molecular bound is especially uniform: arbitrary nuclear arrangements cost only a constant number of excess electrons per nucleus. The atomic results then identify two distinct levels of information. A fixed outer tail has a bounded physical radius; a large but relatively small exterior population follows a universal Thomas–Fermi scaling law.

The full correlated model is the decisive setting. Electron correlations can alter local behavior, so results in a restricted variational model do not automatically carry over. The family claims a mechanism for recovering the relevant screening laws directly from correlated states while preserving the exact leading coefficients.

6. The main idea

The physical mechanism is screening. Electrons nearer the nucleus partly cancel its Coulomb attraction, leaving outer electrons to see a reduced effective field. A Thomas–Fermi description relates local density to the positive part of this screened field $F$:

$$\rho\approx k(F_+)^{3/2},\qquad k=\frac{2^{3/2}}{3\pi^2},$$

where $F_+=\max(F,0)$. The challenge is to justify the appropriate comparisons for a correlated quantum state, rather than assume this relation as the model.

M1 localizes particles while recording the positions and spins removed from each region. The remaining conditional state still carries the correlations relevant to that region. Local trial-electron insertions and lower energy comparisons are arranged with matching semiclassical kinetic coefficients. A coarse kinetic inequality controls density, while the sharp coefficient comes from separate phase-space and eigenvalue-counting comparisons.

Noisy observations then produce conditional densities and screened fields. Conditioning changes probability laws and costs energy; the estimates keep those costs explicit. Positive comparison fields are propagated outward through successive spatial scales, controlling the accumulated errors. For molecules, the geometry is organized so the cost is proportional to the number of nuclei, rather than to the number of scales.

For M2, particle number is first allowed to vary with a price $\lambda>0$:

$$P(\lambda)=\min_{N\geq0}\bigl(E_Z(N)+\lambda N\bigr).$$

A price makes local insertion and deletion compatible with the optimization. At length scale $s=\lambda^{-1/4}$, the missing charge is of order $s^{-3}$. The manuscript identifies its universal coefficient for every minimizing sector, including prices where several sectors tie. Integrating the slopes of this priced energy and bracketing fixed particle numbers recovers the $m^{7/3}$ law without assuming convexity of $E_Z(N)$.

For M3, the limiting neutral screened field has the homogeneous form $F(x)=A|x|^{-4}$. Its Thomas–Fermi density is proportional to $|x|^{-6}$, whose exterior mass is proportional to $r^{-3}$. Solving “exterior mass equals $m$” therefore gives $r$ proportional to $m^{-1/3}$. The substantial work is obtaining this profile from all allowed quantum ground states, without assuming spherical symmetry, and transferring conditional smoothed densities back to the actual expected electron tail.

7. Scope, limitations, and open questions

These claims concern fixed nuclei, nonrelativistic kinetic energy, Coulomb interactions and two spin states. They do not directly describe relativistic heavy atoms, moving nuclei, magnetic fields or finite-temperature states. The universal constants depend on the stated physical normalization even when they are independent of $Z$.

Strict binding must remain distinct from threshold-state existence. The generalized energy theorem requires both a growing number of removed electrons and a much larger nuclear charge. The radius theorem retains its iterated limit order. Neither theorem predicts individual chemical shell oscillations or fixed-$m$ convergence.

The manuscripts provide universal bounds and asymptotic coefficients rather than a numerical table of ionization energies for particular elements. Effective error rates, sharper constants in the uniform bounds, and extensions to other physical Hamiltonians are separate questions. They should not be inferred from the leading laws alone.

8. Sources and further reading

The historical links in section 3 lead to Solovej’s Hartree–Fock theorem and formulation of generalized ionization. Other primary predecessors are Lieb and Simon’s The Thomas–Fermi theory of atoms, molecules and solids and Seco, Sigal and Solovej’s Bound on the ionization energy of large atoms. M1 is the common foundation for the present family; M2 gives the energy limit and its price construction; M3 gives the radius limits and records its foundation inputs explicitly.

Source manuscripts

• M1. OpenAI, Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms (September 24, 2026).
• M2. OpenAI, Generalized ionization energies for full Coulomb atoms (September 24, 2026).
• M3. OpenAI, Generalized outer-electron radii of neutral Coulomb atoms (September 24, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.